Simplify; express your answer in exponential form. Assume $a\neq 0, r\neq 0$. $\dfrac{{ar^{-1}}}{{(a^{-2}r^{-2})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${ar^{-1} = ar^{-1}}$ On the left, we have ${a}$ to the exponent ${1}$ . Now ${1 \times 1 = 1}$ , so ${a = a}$ Apply the ideas above to simplify the equation. $\dfrac{{ar^{-1}}}{{(a^{-2}r^{-2})^{-3}}} = \dfrac{{ar^{-1}}}{{a^{6}r^{6}}}$ Break up the equation by variable and simplify. $\dfrac{{ar^{-1}}}{{a^{6}r^{6}}} = \dfrac{{a}}{{a^{6}}} \cdot \dfrac{{r^{-1}}}{{r^{6}}} = a^{{1} - {6}} \cdot r^{{-1} - {6}} = a^{-5}r^{-7}$